Bohmian Mechanics

Bohmian mechanics, which is also called the de Broglie-Bohm theory,
the pilot-wave model, and the causal interpretation of quantum
mechanics, is a version of quantum theory discovered by Louis de
Broglie in 1927 and rediscovered by David Bohm in 1952. It is the
simplest example of what is often called a hidden variables
interpretation of quantum mechanics. In Bohmian mechanics a system of
particles is described in part by its wave function, evolving, as
usual, according to Schrödinger’s equation. However, the
wave function provides only a partial description of the system. This
description is completed by the specification of the actual positions
of the particles. The latter evolve according to the
“guiding equation”,
which expresses the velocities of the particles in terms of the wave
function. Thus, in Bohmian mechanics the configuration of a system of
particles evolves via a deterministic motion choreographed by the wave
function. In particular, when a particle is sent into a two-slit
apparatus, the slit through which it passes and its location upon
arrival on the photographic plate are completely determined by its
initial position and wave function.

Bohmian mechanics inherits and makes explicit the nonlocality implicit
in the notion, common to just about all formulations and
interpretations of quantum theory, of a wave function on the
configuration space of a many-particle system. It accounts for all of
the phenomena governed by nonrelativistic quantum mechanics, from
spectral lines and scattering theory to superconductivity, the quantum
Hall effect and quantum computing. In particular, the usual
measurement postulates of quantum theory, including collapse of the
wave function and probabilities given by the absolute square of
probability amplitudes, emerge from an analysis of the two equations
of motion: Schrödinger’s equation and the guiding equation.
No invocation of a special, and somewhat obscure, status for
observation is required.

1. The Completeness of the Quantum Mechanical Description

Conceptual difficulties have plagued quantum mechanics since its
inception, despite its extraordinary predictive successes. The basic
problem, plainly put, is this: It is not at all clear what quantum
mechanics is about. What, in fact, does quantum mechanics describe?

It might seem, since it is widely agreed that any quantum mechanical
system is completely described by its wave function, that quantum
mechanics is fundamentally about the behavior of wave functions. Quite
naturally, no physicist wanted this to be true more than did Erwin
Schrödinger, the father of the wave function. Nonetheless,
Schrödinger ultimately found this impossible to believe. His
difficulty had little to do with the novelty of the wave function:

That it is an abstract, unintuitive mathematical construct is a
scruple that almost always surfaces against new aids to thought and
that carries no great message. (Schrödinger [1935] 1980: 327)

Rather, it was that the “blurring” that the spread out
character of the wave function suggests “affects macroscopically
tangible and visible things, for which the term ‘blurring’
seems simply wrong”.

For example, in the same paper Schrödinger noted that it may
happen in radioactive decay that

the emerging particle is described … as a spherical wave
… that impinges continuously on a surrounding luminescent
screen over its full expanse. The screen however does not show a more
or less constant uniform surface glow, but rather lights up at
one instant at one spot …. (Schrödinger
[1935] 1980: 327–328)

And he observed that one can easily arrange, for example by including
a cat in the system, “quite ridiculous cases” with

the \(\psi\)-function of the entire system having in it the living and
the dead cat (pardon the expression) mixed or smeared out in equal
parts. (Schrödinger [1935] 1980: 328)

It is thus because of the “measurement problem”, of
macroscopic superpositions, that Schrödinger found it difficult
to regard the wave function as “representing reality”. But
then what does? With evident disapproval, Schrödinger observes
that

the reigning doctrine rescues itself or us by having recourse to
epistemology. We are told that no distinction is to be made between
the state of a natural object and what I know about it, or perhaps
better, what I can know about it if I go to some trouble.
Actually—so they say—there is intrinsically only
awareness, observation, measurement. (Schrödinger [1935] 1980:
328)

Many physicists pay lip service to the Copenhagen
interpretation—that quantum mechanics is fundamentally about
observation or results of measurement. But it is becoming increasingly
difficult to find any who, when pressed, will defend this
interpretation. It seems clear that quantum mechanics is fundamentally
about atoms and electrons, quarks and strings, not those particular
macroscopic regularities associated with what we call
measurements of the properties of these things. But if these
entities are not somehow identified with the wave function
itself—and if talk of them is not merely shorthand for elaborate
statements about measurements—then where are they to be found in
the quantum description?

There is, perhaps, a very simple reason why it is so difficult to
discern in the quantum description the objects we believe quantum
mechanics ought to describe. Perhaps the quantum mechanical
description is not the whole story, a possibility most prominently
associated with Albert Einstein. (For a general discussion of
Einstein’s scientific philosophy, and in particular of his
approach to the conflicting positions of realism and positivism, see
the entry on
Einstein’s philosophy of science.)

In 1935 Einstein, Boris Podolsky and Nathan Rosen defended this
possibility in their famous EPR paper. They concluded with this
observation:

While we have thus shown that the wave function does not provide a
complete description of the physical reality, we left open the
question of whether or not such a description exists. We believe,
however, that such a theory is possible. (Einstein et al. 1935: 780)

Later, on the basis of more or less the same considerations as those
of Schrödinger quoted above, Einstein again concluded that the
wave function does not provide a complete description of individual
systems, an idea he called “this most nearly obvious
interpretation” (Einstein 1949: 672). In relation to a theory
incorporating a more complete description, Einstein remarked that

the statistical quantum theory would … take an approximately
analogous position to the statistical mechanics within the framework
of classical mechanics. (Einstein 1949: 672)

We note here, and show below, that Bohmian mechanics exactly fits this
description.

2. The Impossibility of Hidden Variables … or the Inevitability of Nonlocality?

John von Neumann, one of the greatest mathematicians of the twentieth
century, claimed that he had proven that Einstein’s dream of a
deterministic completion or reinterpretation of quantum theory was
mathematically impossible. He concluded that

It is therefore not, as is often assumed, a question of a
re-interpretation of quantum mechanics—the present system of
quantum mechanics would have to be objectively false, in order that
another description of the elementary processes than the statistical
one be possible. (von Neumann [1932] 1955: 325)

Physicists and philosophers of science almost universally accepted von
Neumann’s claim. For example, Max Born, who formulated the
statistical interpretation of the wave function, assured us that

No concealed parameters can be introduced with the help of which the
indeterministic description could be transformed into a deterministic
one. Hence if a future theory should be deterministic, it cannot be a
modification of the present one but must be essentially different.
(Born 1949: 109)

Bohmian mechanics is a counterexample to the claims of von Neumann.
Thus von Neumann’s argument must be wrong. In fact, according to
John Bell, von Neumann’s assumptions (about the relationships
among the values of quantum observables that must be satisfied in a
hidden-variables theory) are so unreasonable that the “the proof
of von Neumann is not merely false but foolish!”
(Mermin 1993: 805, fn 8, quoting an interview in Omni, May,
1988: 88). Nonetheless, some physicists continue to rely on von
Neumann’s proof.

Recently, however, physicists more commonly cite the
Kochen-Specker Theorem
and, more frequently,
Bell’s inequality
in support of the contention that a deterministic completion of
quantum theory is impossible. We still find, a quarter of a century
after the rediscovery of Bohmian mechanics in 1952, statements such as
these:

The proof he [von Neumann] published …, though it was made much
more convincing later on by Kochen and Specker (1967), still uses
assumptions which, in my opinion, can quite reasonably be questioned.
… In my opinion, the most convincing argument against the
theory of hidden variables was presented by J.S. Bell (1964). (Wigner
[1976] 1983: 291)

Now there are many more statements of a similar character that we
could cite. This quotation is significant because Wigner was one of
the leading physicists of his generation. Unlike most of his
contemporaries, moreover, he was also profoundly concerned about the
conceptual foundations of quantum mechanics and wrote on the subject
with great clarity and insight.

There was, however, one physicist who wrote on this subject with even
greater clarity and insight than Wigner himself: the very J. S. Bell
whom Wigner praises for demonstrating the impossibility of a
deterministic completion of quantum theory such as Bohmian mechanics.
Here’s how Bell himself reacted to Bohm’s discovery:

But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be
introduced, into nonrelativistic wave mechanics, with the help of
which the indeterministic description could be transformed into a
deterministic one. More importantly, in my opinion, the subjectivity
of the orthodox version, the necessary reference to the
“observer”, could be eliminated. …

But why then had Born not told me of this “pilot wave”? If
only to point out what was wrong with it? Why did von Neumann not
consider it? More extraordinarily, why did people go on producing
“impossibility” proofs, after 1952, and as recently as
1978? … Why is the pilot wave picture ignored in text books?
Should it not be taught, not as the only way, but as an antidote to
the prevailing complacency? To show us that vagueness, subjectivity,
and indeterminism, are not forced on us by experimental facts, but by
deliberate theoretical choice? (Bell 1982, reprinted in 1987c: 160)

Wigner to the contrary notwithstanding, Bell did not establish the
impossibility of a deterministic reformulation of quantum theory, nor
did he ever claim to have done so. On the contrary, until his untimely
death in 1990, Bell was the prime proponent, and for much of this
period almost the sole proponent, of the very theory, Bohmian
mechanics, that he supposedly demolished.

Bohmian mechanics is of course as much a counterexample to the
Kochen-Specker argument for the impossibility of hidden variables as
it is to the one of von Neumann. It is obviously a counterexample to
any such argument. However reasonable the assumptions of such an
argument, some of them must fail for Bohmian mechanics.

Wigner was quite right to suggest that the assumptions of Kochen and
Specker are more convincing than those of von Neumann. They appear, in
fact, to be quite reasonable indeed. However, they are not. The
impression that they are arises from a pervasive error, an uncritical
realism about operators, that we discuss below in the sections on
quantum observables,
spin, and
contextuality.

John Bell replaced the “arbitrary axioms” (Bell 1966,
reprinted 1987c: 11) of Kochen-Specker and others by an assumption of
locality, of no action-at-a-distance. It would be hard to argue
against the reasonableness of such an assumption, even if one were so
bold as to doubt its inevitability. Bell showed that any
hidden-variables formulation of quantum mechanics must be nonlocal,
as, indeed, Bohmian mechanics is. But he showed much much more. (For
more detail on Bell’s locality assumption, see Goldstein et al.
2011 and Norsen 2011.)

In a celebrated paper he published in 1964, Bell showed that quantum
theory itself is irreducibly nonlocal. (More precisely, Bell’s
analysis applies to any single-world version of quantum theory, i.e.,
any version for which measurements have outcomes that, while they may
be random, are nonetheless unambiguous and definite, in contrast to
the situation with Everett’s many-worlds version of quantum
theory.) This fact about quantum mechanics, based as it is on a short
and mathematically simple analysis, could have been recognized soon
after the discovery of quantum theory in the 1920’s. That this
did not happen is no doubt due in part to the obscurity of
orthodox quantum theory
and to the ambiguity of its commitments. (It almost did happen:
Schrödinger 1935, in his famous cat paper, came remarkably close
to discovering a Bell-type argument for quantum nonlocality. For
details see Hemmick and Shakur 2012, chapter 4.) It was, in fact, his
examination of Bohmian mechanics that led Bell to his nonlocality
analysis. In the course of investigating Bohmian mechanics, he
observed that:

in this theory an explicit causal mechanism exists whereby the
disposition of one piece of apparatus affects the results obtained
with a distant piece. …

Bohm of course was well aware of these features of his scheme, and has
given them much attention. However, it must be stressed that, to the
present writer’s knowledge, there is no proof that
any hidden variable account of quantum mechanics
must have this extraordinary character. It would therefore be
interesting, perhaps, to pursue some further “impossibility
proofs”, replacing the arbitrary axioms objected to above by
some condition of locality, or of separability of distant systems.
(Bell 1966: 452; reprinted 1987c: 11)

In a footnote, Bell added that “Since the completion of this
paper such a proof has been found” (1966: 452, fn. 19). He
published it in his 1964 paper, “On the Einstein-Podolsky-Rosen
Paradox”. In this paper he derives Bell’s inequality, the
basis of his conclusion of quantum nonlocality. (See the entry on
Bell’s Theorem.
For a discussion of how nonlocality emerges in Bohmian mechanics, see
Section 13.)

It is worth stressing that Bell’s analysis indeed shows that any
(single-world) account of quantum phenomena must be nonlocal, not just
any hidden variables account. Bell showed that the predictions of
standard quantum theory itself imply nonlocality. Thus if these
predictions govern nature, then nature is nonlocal. [That nature is so
governed, even in the crucial EPR-correlation experiments, has by now
been established by a great many experiments. The first rather
conclusive such experiment was that of Aspect (Aspect, Dalibard, &
Zanghì 1982). More conclusive still is the experiment of Weihs
et al. 1998. Very recently there have been several “loop-hole
free” tests of Bell’s inequality (Giustina et al. 2015;
Hensen et al. 2015; and Shalm et al. 2015).]

It is important to note that to the limited degree to which
determinism plays a role in the EPR argument, it is not
assumed but inferred. What is held sacred is the principle of
“local causality”—or “no action at a
distance”…

It is remarkably difficult to get this point across, that determinism
is not a presupposition of the analysis. (Bell 1981a,
reprinted 1987c: 143)

Despite my insistence that the determinism was inferred rather than
assumed, you might still suspect somehow that it is a preoccupation
with determinism that creates the problem. Note well then that the
following argument makes no mention whatever of determinism. …
Finally you might suspect that the very notion of particle, and
particle orbit … has somehow led us astray. … So the
following argument will not mention particles, nor indeed fields, nor
any other particular picture of what goes on at the microscopic level.
Nor will it involve any use of the words “quantum mechanical
system”, which can have an unfortunate effect on the discussion.
The difficulty is not created by any such picture or any such
terminology. It is created by the predictions about the correlations
in the visible outputs of certain conceivable experimental set-ups.
(Bell 1981a, reprinted 1987c: 150)

The “problem” and “difficulty” to which Bell
refers above is the conflict between the predictions of quantum theory
and what can be inferred, call it \(C\), from an assumption of
locality in Bohm’s version of the EPR argument, a conflict
established by Bell’s inequality. \(C\) happens to concern the
existence of a certain kind of hidden variables, what might be called
local hidden variables, but this fact is of little substantive
importance. What is important is not so much the identity of \(C\) as
the fact that \(C\) is incompatible with the predictions of quantum
theory. The identity of \(C\) is, however, of great historical
significance: it is responsible for the misconception that Bell proved
that hidden variables are impossible, a belief that physicists until
recently almost universally shared, as well as for the view, even now
almost universally held, that what Bell’s result does is to rule
out local hidden variables, a view that is misleading.

Here again is Bell, expressing the logic of his two-part
demonstration of quantum nonlocality, the first part of which is
Bohm’s version of the EPR argument, concerning EPRB correlations:

Let me summarize once again the logic that leads to the impasse. The
EPRB correlations are such that the result of the experiment on one
side immediately foretells that on the other, whenever the analyzers
happen to be parallel. If we do not accept the intervention on one
side as a causal influence on the other, we seem obliged to admit that
the results on both sides are determined in advance anyway,
independently of the intervention on the other side, by signals from
the source and by the local magnet setting. But this has implications
for non-parallel settings which conflict with those of quantum
mechanics. So we cannot dismiss intervention on one side as a
causal influence on the other. (Bell 1981a, reprinted 1987c: 149)

As with just about everything else in the foundations of quantum
mechanics, there remains considerable controversy about what exactly
Bell’s analysis demonstrates. (For further insight into the
various controversies see Maudlin 2014 and Goldstein et al. 2011.)
Nonetheless, the opinion of Bell himself about what he showed is
perfectly clear. See Norsen 2011 for a nice overview of Bell’s
views on the matter.

3. History

The pilot-wave approach to quantum theory was initiated by Einstein,
even before the discovery of quantum mechanics itself. Einstein hoped
that interference phenomena involving particle-like photons could be
explained if the motion of the photons was somehow guided by the
electromagnetic field—which would thus play the role of what he
called a Führungsfeld or guiding field (see Wigner
[1976] 1983: 262 and Bacciagaluppi and Valentini 2009: Ch. 9). While
the notion of the electromagnetic field as guiding field turned out to
be rather problematical, Max Born explored the possibility that the
wave function could play this role, of guiding field or pilot wave,
for a system of electrons in his early paper founding quantum
scattering theory (Born 1926). Heisenberg was profoundly
unsympathetic.

Not long after Schrödinger’s discovery of wave mechanics in
1926, i.e., of Schrödinger’s equation, Louis de Broglie in
effect discovered Bohmian mechanics: In 1927, de Broglie found an
equation of particle motion equivalent to
the guiding equation
for a scalar wave function (de Broglie 1928: 119), and he explained
at the 1927 Solvay Congress how this motion could account for quantum
interference phenomena. However, despite what is suggested by
Bacciagaluppi and Valentini (2009), de Broglie responded very poorly
to an objection of Wolfgang Pauli (Pauli 1928) concerning inelastic
scattering, no doubt making a rather bad impression on the illustrious
audience at the congress.

Born and de Broglie very quickly abandoned the pilot-wave approach and
became enthusiastic supporters of the rapidly developing consensus in
favor of the Copenhagen interpretation. David Bohm (1952) rediscovered
de Broglie’s pilot-wave theory in 1952. He was the first person
to genuinely understand its significance and implications. John Bell
became its principal proponent during the sixties, seventies and
eighties.

For a very good discussion of the history of quantum mechanics, the
debates about its foundations, and about the reception of Bohmian
mechanics in particular, see Bricmont 2016. See also Beller 1999.

4. The Defining Equations of Bohmian Mechanics

In Bohmian mechanics the wave function, obeying
Schrödinger’s equation, does not provide a complete
description or representation of a quantum system. Rather, it governs
the motion of the fundamental variables, the positions of the
particles: In the Bohmian mechanical version of nonrelativistic
quantum theory, quantum mechanics is fundamentally about the behavior
of particles; the particles are described by their positions, and
Bohmian mechanics prescribes how these change with time. In this
sense, for Bohmian mechanics the particles are primary, or primitive,
while the wave function is secondary, or derivative.

Warning: It is the positions of the particles in Bohmian mechanics
that are its “hidden variables”, an unfortunate bit of
terminology. As Bell writes, referring to Bohmian mechanics and
similar theories,

Absurdly, such theories are known as “hidden variable”
theories. Absurdly, for there it is not in the wavefunction that one
finds an image of the visible world, and the results of experiments,
but in the complementary “hidden”(!) variables. Of course
the extra variables are not confined to the visible
“macroscopic” scale. For no sharp definition of such a
scale could be made. The “microscopic” aspect of the
complementary variables is indeed hidden from us. But to admit things
not visible to the gross creatures that we are is, in my opinion, to
show a decent humility, and not just a lamentable addiction to
metaphysics. In any case, the most hidden of all variables, in the
pilot wave picture, is the wavefunction, which manifests itself to us
only by its influence on the complementary variables. (1987a,
reprinted 1987c: 201–202)

Bohmian mechanics is the minimal completion of
Schrödinger’s equation, for a nonrelativistic system of
particles, to a theory describing a genuine motion of particles. For
Bohmian mechanics the state of a system of \(N\) particles is
described by its wave function \(\psi = \psi(\boldsymbol{q}_1 ,\ldots
,\boldsymbol{q}_N) = \psi(q)\), a complex (or spinor-valued) function
on the space of possible configurations \(q\) of the system, together
with its actual configuration \(Q\) defined by the actual positions
\(\mathbf{Q}_1 ,\ldots ,\mathbf{Q}_N\) of its particles. (The word
“spinor” refers to a suitable array of complex numbers in
place of a single one. Spinor-valued wave functions are used in
quantum mechanics to describe electrons and other quantum particles
that “have spin”.) The theory is then defined by two
evolution equations: Schrödinger’s equation
\[
i\hslash\frac{\partial \psi}{\partial t} = H\psi
\]
for
\(\psi(t)\), where \(H\) is the nonrelativistic (Schrödinger)
Hamiltonian, containing the masses of the particles and a potential
energy term, and (writing \(\Im[z]\) for the imaginary part \(b\) of a
complex number \(z = a +\di b)\) a first-order evolution equation,

for \(Q(t)\), the simplest first-order evolution equation for the
positions of the particles that is compatible with the Galilean (and
time-reversal) covariance of the Schrödinger evolution
(Dürr, Goldstein, & Zanghì 1992a: 852–854). Here
\(\hslash\) is Planck’s constant divided by \(2\pi\),
m\(_k\) is the mass of the \(k\)-th particle, and
\(\partial_k = (\partial /\partial x_k,\partial /\partial y_k,\partial
/\partial z_k)\) is the gradient with respect to the generic
coordinates \(\boldsymbol{q}_k = (x_k,y_k,z_k)\) of the \(k\)-th
particle. If \(\psi\) is spinor-valued, the two products involving
\(\psi\) in the equation should be understood as scalar products
(involving sums of products of spinor components). When external
magnetic fields are present, the gradient should be understood as the
covariant derivative, involving the vector potential. (Since the
denominator on the right hand side of the guiding equation vanishes at
the nodes of \(\psi\), global existence and uniqueness for the Bohmian
dynamics is a nontrivial matter. It is proven in Berndl, Dürr, et
al. 1995 and in Teufel and Tumulka 2005.)

For an \(N\)-particle system these two equations (together with the
detailed specification of the Hamiltonian, including all interactions
contributing to the potential energy) completely define Bohmian
mechanics. This deterministic theory of particles in motion accounts
for all the phenomena of nonrelativistic quantum mechanics, from
interference effects to spectral lines (Bohm 1952: 175–178) to
spin (Bell 1964: 10). It does so in an entirely ordinary manner, as we
explain in the following sections.

For a scalar wave function, describing particles without spin, the
form of
the guiding equation
above is a little more complicated than necessary, since the complex
conjugate of the wave function, which appears in the numerator and the
denominator, cancels out. If one looks for an evolution equation for
the configuration compatible with the space-time symmetries of
Schrödinger’s equation, one almost immediately arrives at
the guiding equation in this simpler form as the simplest
possibility.

However, the form above has two advantages: First, it makes sense for
particles with spin—and, in fact, Bohmian mechanics without
further ado accounts for all the apparently paradoxical quantum
phenomena associated with spin. Secondly, and this is crucial to the
fact that Bohmian mechanics is empirically equivalent to orthodox
quantum theory, the right hand side of the guiding equation is
\(J/\varrho\), the ratio of the quantum probability current to the
quantum probability density. This shows that it should require no
imagination whatsoever to guess the guiding equation from
Schrödinger’s equation, provided one is looking for one,
since the classical formula for current is density times velocity.
Moreover, it follows from the quantum continuity equation \(\partial
\varrho /\partial t + \textrm{div} J = 0\), an immediate consequence
of Schrödinger’s equation, that if at some time (say the
initial time) the configuration \(Q\) of our system is random, with
distribution given by \(\lvert \psi \rvert^2 = \psi *\psi\), this will
always be true (provided the system does not interact with its
environment).

This demonstrates that it is wrong to claim that the predictions of
quantum theory are incompatible with the existence of hidden
variables, with an underlying deterministic model in which quantum
randomness arises from averaging over ignorance. Bohmian mechanics
provides us with just such a model: For any quantum experiment we
merely take as the relevant Bohmian system the combined system,
including the system upon which the experiment is performed as well as
all the measuring instruments and other devices used to perform the
experiment (together with all other systems with which these have
significant interaction over the course of the experiment). We then
obtain the “hidden variables” model by regarding the
initial configuration of this big system as random in the usual
quantum mechanical way, with distribution given by \(\lvert \psi
\rvert^2\). The guiding equation for the big system then transforms
the initial configuration into the final configuration at the
conclusion of the experiment. It then follows that this final
configuration of the big system, including in particular the
orientation of instrument pointers, will also be distributed in the
quantum mechanical way. Thus our deterministic Bohmian model yields
the usual quantum predictions for the results of the experiment.

As the preceding paragraph suggests, and as we discuss in more detail
later, Bohmian mechanics does not need any “measurement
postulates” or axioms governing the behavior of other
“observables”. Any such axioms would be at best redundant
and could be inconsistent.

Besides the
guiding equation,
there are other velocity formulas with nice properties, including
Galilean symmetry, and yielding theories that are empirically
equivalent to orthodox quantum theory—and to Bohmian mechanics
(Deotto & Ghirardi 1998). The Bohmian choice is arguably the
simplest. Moreover, Wiseman (2007) has shown that it is the Bohmian
velocity formula, given by the
guiding equation,
that, according to orthodox quantum theory, would be found in a
“weak measurement” of the velocity of a particle. (Marian,
Zanghì, and Oriols 2016 have recently proposed an explicit
procedure for performing such a measurement.) And, somewhat
paradoxically, it can be shown (Dürr, Goldstein, &
Zanghì 2009) that according to Bohmian mechanics such a
measurement is indeed a genuine measurement of the particle’s
velocity—despite the existence of empirically equivalent
velocity formulas! Similarly, weak measurements could be used to
measure trajectories. In fact, quite recently Kocsis et al. (2011)
have used weak measurements to reconstruct the trajectories for single
photons “as they undergo two-slit interference”, finding
“those predicted in the Bohm-de Broglie interpretation of
quantum mechanics”. And Mahler et al. 2016 have experimentally
found, using weak measurements, “Bohmian trajectories” for
entangled photons, illustrating quantum nonlocality and the phenomenon
of “surreal Bohmian trajectories”.

For a single particle the guiding equation defines the motion of a
particle guided by a wave in physical 3-dimensional space. One might
expect that similar motions might arise classically. Couder & Fort
(2006) have shown, by investigating interference-like phenomena in the
motion of bouncing oil droplets in a fluid, that this is indeed so.
Bush (2015) has further explored this sort of possibility for the
emergence of a Bohmian version of quantum mechanics from something
like classical fluid dynamics. A serious obstacle to the success of
such a program is the quantum entanglement and nonlocality
characteristic of many-particle quantum systems.

The “many interacting worlds” approach of Hall, Deckert,
& Wiseman (2014), independently advanced by Sebens (2015), retains
particle trajectories but attempts to do away with a wave-function at
the level of fundamental ontology. What takes its place is a large
number of trajectories of points in configuration space, each
corresponding to the motions of a finite number of particles in
physical space regarded as describing a a world in its own right.
These interact in such a way as to mimic trajectories guided by a wave
function: a coarse-grained density of worlds takes the place of
\(\lvert \psi \rvert^2\). The velocities of the world-particles are
required to at least approximately be related to their configurations
via some smooth function on configuration space. This condition would
be a rather surprising one for the sorts of large systems studied in
statistical mechanics. Even more unmotivated from a classical
perspective, the velocity function must be the gradient of a
multivalued function that obeys a Bohr-Sommerfeld-like quantum
condition.

We stress that Bohmian mechanics should be regarded as a theory in its
own right. Its viability does not depend on its being derivable from
some other theory, classical or otherwise.

5. The Quantum Potential

Bohmian mechanics as presented here is a first-order theory, in which
it is the velocity, the rate of change of position, that is
fundamental. It is this quantity, given by
the guiding equation,
that the theory specifies directly and simply. The second-order
(Newtonian) concepts of acceleration and force, work and energy do not
play any fundamental role. Bohm, however, did not regard his theory in
this way. He regarded it, fundamentally, as a second-order theory,
describing particles moving under the influence of forces, among
which, however, is a force stemming from a “quantum
potential”.

In his 1952 hidden-variables paper, Bohm arrived at his theory by
writing the wave function in the polar form \(\psi =
R\)exp\((iS/\hslash)\), where \(S\) and \(R\) are real, with \(R\)
nonnegative, and rewriting Schrödinger’s equation in terms
of these new variables to obtain a pair of coupled evolution
equations: the continuity equation for \(\varrho = R^2\) and a
modified Hamilton-Jacobi equation for \(S\). This differs from the
usual classical Hamilton-Jacobi equation only by the appearance of an
extra term, the quantum potential
\[
U = {-}\sum_k (\hslash^2 /2m_k) (\partial_{k}^2 R / R ),
\]
alongside the
classical potential energy term.

Bohm then used the modified Hamilton-Jacobi equation to define
particle trajectories just as one does for the classical
Hamilton-Jacobi equation, that is, by identifying \(\partial_k S\)
with \(m_k\boldsymbol{v}_k\), i.e., by setting
\[
d\mathbf{Q}_k /dt = \partial_k S / m_k.
\]
This is
equivalent to
the guiding equation
for particles without spin. [In this form the (pre-Schrödinger
equation) de Broglie relation \(\boldsymbol{p} = \hslash
\boldsymbol{k}\), as well as by the eikonal equation of classical
optics, already suggest the guiding equation.] The resulting motion is
precisely what would be obtained classically if the particles were
acted upon by the force generated by the quantum potential, in
addition to the usual forces.

The quantum potential formulation of the de Broglie-Bohm theory is
still fairly widely used. For example, the monographs by Bohm and
Hiley and by Holland present the theory in this way. And regardless of
whether or not we regard the quantum potential as fundamental, it can
in fact be quite useful. In order to see most clearly that Newtonian
mechanics should be expected to emerge from Bohmian mechanics in the
classical limit, it is convenient to transform the theory into
Bohm’s Hamilton-Jacobi form. Then the (size of the) quantum
potential provides a measure of the deviation of Bohmian mechanics
from its classical approximation. Moreover, the quantum potential is
also useful for developing approximation schemes for solutions to
Schrödinger’s equation (Nerukh & Frederick 2000).

However, Bohm’s rewriting of Schrödinger’s equation
in terms of variables that seem interpretable in classical terms is
not without a cost. The most obvious is an increase in complexity:
Schrödinger’s equation is rather simple, and it is linear,
whereas the modified Hamilton-Jacobi equation is somewhat complicated,
and highly nonlinear. Moreover the latter, since it involves \(R\),
requires the continuity equation for its closure. The quantum
potential itself is neither simple nor natural. Even to Bohm it seemed
“rather strange and arbitrary” (Bohm 1980: 80). And it is
not very satisfying to think of the quantum revolution as amounting to
the insight that nature is classical after all, except that there is
in nature what appears to be a rather ad hoc additional force
term, the one arising from the quantum potential. The artificiality
that the quantum potential suggests is the price one pays for casting
a highly nonclassical theory into a classical mold.

Moreover, the connection between classical mechanics and Bohmian
mechanics that the quantum potential suggests is rather misleading.
Bohmian mechanics is not simply classical mechanics with an additional
force term. In Bohmian mechanics the velocities are not independent of
positions, as they are classically, but are constrained by the guiding
equation. (In classical Hamilton-Jacobi theory we also have this
equation for the velocity, but there the Hamilton-Jacobi function
\(S\) can be entirely eliminated and the description in terms of \(S\)
simplified and reduced to a finite-dimensional description, with basic
variables the positions and the (unconstrained) momenta of all the
particles, given by Hamilton’s or Newton’s equations.)

Arguably, the most serious flaw in the quantum potential formulation
of Bohmian mechanics is that it gives a completely false impression of
the lengths to which we must go in order to convert orthodox quantum
theory into something more rational. The quantum potential suggests,
as has often been stated, that transforming Schrödinger’s
equation into a theory that can account in “realistic”
terms for quantum phenomena, many of which are dramatically nonlocal,
requires adding to the theory a complicated quantum potential of a
grossly nonlocal character. It should be clear that this view is
inappropriate. After all, the quantum potential need not even be
mentioned in the formulation of Bohmian mechanics, and it in
any case merely reflects the wave function, which Bohmian mechanics
shares with orthodox quantum theory.

6. The Two-Slit Experiment

According to Richard Feynman, the two-slit experiment for electrons is

a phenomenon which is impossible, absolutely impossible, to
explain in any classical way, and which has in it the heart of quantum
mechanics. In reality it contains the only mystery. (Feynman,
Leighton, & Sands 1963: 37–2)

This experiment

has been designed to contain all of the mystery of quantum mechanics,
to put you up against the paradoxes and mysteries and peculiarities of
nature one hundred per cent. (Feynman 1967: 130)

As to the question,

How does it really work? What machinery is actually producing this
thing? Nobody knows any machinery. Nobody can give you a deeper
explanation of this phenomenon than I have given; that is, a
description of it. (Feynman 1967: 145)

But Bohmian mechanics is just such a deeper explanation. It resolves
in a rather straightforward manner the dilemma of the appearance of
both particle and wave properties in one and the same phenomenon:
Bohmian mechanics is a theory of motion describing a particle (or
particles) guided by a wave. Here we have a family of Bohmian
trajectories for the two-slit experiment.

While each trajectory passes through only one slit, the wave passes
through both; the interference profile that therefore develops in the
wave generates a similar pattern in the trajectories guided by the
wave.

Compare Feynman’s presentation with Bell’s:

Is it not clear from the smallness of the scintillation on the screen
that we have to do with a particle? And is it not clear, from the
diffraction and interference patterns, that the motion of the particle
is directed by a wave? De Broglie showed in detail how the motion of a
particle, passing through just one of two holes in screen, could be
influenced by waves propagating through both holes. And so influenced
that the particle does not go where the waves cancel out, but is
attracted to where they cooperate. This idea seems to me so natural
and simple, to resolve the wave-particle dilemma in such a clear and
ordinary way, that it is a great mystery to me that it was so
generally ignored. (Bell [1989] 1987c: 191)

Perhaps the most puzzling aspect of the two-slit experiment is the
following: If, by any means whatsoever, it is possible to determine
the slit through which the particle passes, the interference pattern
will be destroyed. This dramatic effect of observation is, in fact, a
simple consequence of Bohmian mechanics. To see this, one must
consider the meaning of determining the slit through which the
particle passes. This must involve interaction with another system
that the Bohmian mechanical analysis must include.

The destruction of interference is related, naturally enough, to the
Bohmian mechanical analysis of quantum measurement (Bohm 1952). It
occurs via the mechanism that in Bohmian mechanics leads to the
“collapse of the wave function”.

For an accessible presentation of the behavior of Bohmian trajectories
in scattering and tunneling phenomena, see Norsen 2013.

7. The Measurement Problem

The
measurement problem
is the most commonly cited of the conceptual difficulties that plague
quantum mechanics. (It amounts, more or less, to the paradox of
Schrödinger’s cat.) Indeed, for many physicists the
measurement problem is not merely one conceptual difficulty of quantum
mechanics; it is the conceptual difficulty.

The problem is as follows. Suppose that the wave function of any
individual system provides a complete description of that system. When
we analyze the process of measurement in quantum mechanical terms, we
find that the after-measurement wave function for system and apparatus
that arises from Schrödinger’s equation for the composite
system typically involves a superposition over terms corresponding to
what we would like to regard as the various possible results of the
measurement—e.g., different pointer orientations. In this
description of the after-measurement situation it is difficult to
discern the actual result of the measurement—e.g., some specific
pointer orientation. But the whole point of quantum theory, and the
reason we should believe in it, is that it is supposed to provide a
compelling, or at least an efficient, account of our observations,
that is, of the outcomes of measurements. In short, the measurement
problem is this: Quantum theory implies that measurements typically
fail to have outcomes of the sort the theory was created to
explain.

In contrast, if we, like Einstein, regard the description provided by
the wave function as incomplete, the measurement problem vanishes:
There is no measurement problem with a theory or interpretation in
which, as in Bohmian mechanics, the description of the
after-measurement situation includes, in addition to the wave
function, at least the values of the variables that register the
result. In Bohmian mechanics pointers always point.

Often, the measurement problem is expressed a little differently.
Textbook quantum theory provides two rules for the evolution of the
wave function of a quantum system: A deterministic dynamics given by
Schrödinger’s equation when the system is not being
“measured” or observed, and a random collapse of the wave
function to an eigenstate of the “measured
observable” when it is. However, the objection continues,
textbook quantum theory does not explain how to reconcile these two
apparently incompatible rules.

That this formulation of the measurement problem and the preceding one
are more or less equivalent should be reasonably clear: If a wave
function provides a complete description of the after-measurement
situation, the outcome of the measurement must correspond to a wave
function that describes the actual result, that is, a
“collapsed” wave function. Hence the collapse rule. But it
is difficult to take seriously the idea that different laws than those
governing all other interactions should govern those interactions
between system and apparatus that we happen to call measurements.
Hence the apparent incompatibility of the two rules.

The second formulation of the measurement problem, though basically
equivalent to the first, raises an important question: Can Bohmian
mechanics itself reconcile these two dynamical rules? How does Bohmian
mechanics justify the use of the “collapsed” wave function
instead of the original one? This question was answered in
Bohm’s first papers on Bohmian mechanics (Bohm 1952: Part I,
Section 7 and Part II, Section 2). What would nowadays be called
effects of decoherence, which interaction with the environment (air
molecules, cosmic rays, internal microscopic degrees of freedom, etc.)
produces, make difficult the development of significant overlap
between the component of the after-measurement wave function
corresponding to the actual result of the measurement and the other
components of the after-measurement wave function. (This overlap
refers to the configuration space of the very large system that
includes all systems with which the original system and apparatus come
into interaction.) But without such overlap that component all by
itself generates to a high degree of accuracy the future evolution of
the configuration of the system and apparatus. The replacement is thus
justified as a practical matter (see also Dürr, Goldstein &
Zanghì 1992a: Section 5).

Many proponents of orthodox quantum theory believe that decoherence
somehow resolves the measurement problem itself. It is not easy to
understand this belief. In the first formulation of the measurement
problem, nothing prevents us from including in the apparatus all
sources of decoherence. But then decoherence can no longer be in any
way relevant to the argument. Be that as it may, Bohm (1952) gave one
of the best descriptions of the mechanisms of decoherence, though he
did not use the word itself. He recognized its importance several
decades before it became fashionable. (See also the encyclopedia entry
on
The Role of Decoherence in Quantum Mechanics.)

8. The Collapse of the Wave Function

In the previous section we indicated that collapse of the wave
function can be regarded in Bohmian mechanics as a pragmatic affair.
However, there is a sense in which the collapse of the wave function
in Bohmian mechanics is more than a matter of convenience. If we focus
on the appropriate notion of the wave function, not of the composite
of system and apparatus—which strictly speaking remains a
superposition if the composite is treated as closed during the
measurement process—but of the system itself, we find that for
Bohmian mechanics this does indeed collapse, precisely as the quantum
formalism says. The key element here is the notion of the
conditional wave function of a subsystem of a larger system,
which we describe briefly in this section and that Dürr,
Goldstein, & Zanghì 1992a: Section 5, discuss in some
detail, together with the related notion of the effective wave
function.

For the evolution of the wave function, Bohmian mechanics is
formulated in terms of Schrödinger’s equation alone.
Nonetheless the textbook collapse rule is a consequence of the Bohmian
dynamics. To appreciate this one should first note that, since
observation implies interaction, a system under observation cannot be
a closed system but rather must be a subsystem of a larger closed
system, which we may take to be the entire universe, or any smaller
more or less closed system that contains the system to be observed,
the subsystem. The configuration \(Q\) of this larger system
naturally splits into \(X\), the configuration of the subsystem, and
\(Y\), the configuration of the environment of the
subsystem.

Suppose the larger system has wave function \(\Psi = \Psi(q) = \Psi(x,
y)\). According to Bohmian mechanics, the larger system is then
completely described by \(\Psi\), evolving according to
Schrödinger’s equation, together with \(X\) and \(Y\). The
question then arises—and it is a critical question—as to
what should be meant by the wave function of the subsystem.

There is a rather obvious answer for this, a natural function of \(x\)
that suitably incorporates the objective structure at hand, namely the
conditional wave function
\[\psi(x) = \Psi(x, Y)\]
obtained by plugging
the actual configuration of the environment into the wave function of
the larger system. (This definition is appropriate only for scalar
wave functions; for particles with spin the situation would be a
little more complicated.) It then follows immediately that the
configuration of the subsystem obeys
the guiding equation
with the conditional wave function on its right-hand side.

Moreover, taking into account the way that the conditional wave
function depends upon time \(t\)
\[\psi_t (x) = \Psi_t (x, Y_t)\]
via the time dependence
of \(Y\) as well as that of \(\Psi\), it is not difficult to see
(Dürr, Goldstein, & Zanghì 1992a) the following two
things about the evolution of the conditional wave: First, that it
obeys Schrödinger’s equation for the subsystem when that
system is suitably decoupled from its environment. Part of what is
meant by this decoupling is that \(\Psi\) has a special form, what
might be called an effective product form (similar to but more general
than the superposition produced in an “ideal quantum
measurement”), in which case the conditional wave function of
the subsystem is also called its effective wave function.
Second, using the
quantum equilibrium hypothesis,
that it randomly collapses according to the usual quantum mechanical
rules under precisely those conditions on the interaction between the
subsystem and its environment that define an ideal quantum
measurement.

It is perhaps worth noting that orthodox quantum theory lacks the
resources that make it possible to define the conditional wave
function, namely, the actual configuration \(Y\) of the environment.
Indeed, from an orthodox point of view what should be meant by the
wave function of a subsystem is entirely obscure.

9. Quantum Randomness

According to the quantum formalism, for a system with wave function
\(\psi\) the probability density for finding its configuration to be
\(q\) is \(\lvert\psi(q)\rvert^2\). To the extent that the results of
measurement are registered configurationally, at least potentially, it
follows that the predictions of Bohmian mechanics for the results of
measurement must agree with those of orthodox quantum theory (assuming
the same Schrödinger equation for both) provided that it is
somehow true for Bohmian mechanics that configurations are random,
with distribution given by the quantum equilibrium
distribution \(\lvert\psi(q)\rvert^2\). Now the status and
justification of this quantum equilibrium hypothesis is a
rather delicate matter, one that has been explored in considerable
detail (Dürr, Goldstein, & Zanghì 1992a). Here are a
few relevant points.

It is nowadays a rather familiar fact that dynamical systems quite
generally give rise to behavior of a statistical character, with the
statistics given by the (or a) stationary probability distribution for
the dynamics. So it is with Bohmian mechanics, except that for the
Bohmian system stationarity is not quite the right concept. Rather it
is the notion of equivariance that is relevant. A probability
distribution on configuration space \(\varrho^{\psi}\), depending upon
the wave function \(\psi\), is equivariant if
\[(\varrho^{\psi})_t = \varrho^{\psi_t}\]
where the dependence on \(t\) on the right arises from
Schrödinger’s equation and on the left from the evolution
on probability distributions arising from the flow that
the guiding equation
induces. Thus equivariance expresses the mutual compatibility,
relative to \(\varrho^{\psi}\), of the Schrödinger evolution of
the wave function and the Bohmian motion of the configuration. It is
an immediate consequence of
the guiding equation
and the quantum continuity equation that \(\varrho^{\psi} =
\lvert\psi(q)\rvert^2\) is equivariant. (It can be shown in fact that
this is more or less the only equivariant possibility that is suitably
local (Goldstein & Struyve 2007).)

In trying to understand the status in Bohmian mechanics of the quantum
equilibrium distribution, it is perhaps helpful to think of

quantum equilibrium, \(\varrho = \lvert\psi \rvert^2\)

as roughly analogous to (classical)

thermodynamic equilibrium, \(\varrho = \textrm{exp} (-H/kT)
/Z\),

the probability distribution of the phase-space point of a system in
equilibrium at temperature \(T\). (\(Z\) is a normalization constant
called the partition function and \(k\) is Boltzmann’s
constant.) This analogy has several facets: In both cases the
probability distributions are naturally associated with their
respective dynamical systems. In particular, these distributions are
stationary or, what amounts to the same thing within the framework of
Bohmian mechanics, equivariant. In both cases it seems natural to try
to justify these equilibrium distributions by means of mixing-type,
convergence-to-equilibrium arguments (Bohm 1953; Valentini &
Westman 2005). It has been argued, however, that in both cases the
ultimate justification for these probability distributions must be in
terms of statistical patterns that ensembles of actual subsystems
within a typical individual universe exhibit (Bell 1981b,
reprinted 1987c: 129; Dürr, Goldstein, & Zanghì
1992a). In both cases the status of, and justification for,
equilibrium distributions is still controversial. It is also perhaps
worth noting that the typicality-grounded account of quantum
randomness in Bohmian mechanics is extremely similar to
Everett’s account (Everett 1957) of quantum randomness for
“many worlds”, despite the huge metaphysical differences
that exist between these two versions of quantum theory. It can be
shown (Dürr, Goldstein, & Zanghì 1992a) that
probabilities for positions given by the quantum equilibrium
distribution emerge naturally from an analysis of
“equilibrium” for the deterministic dynamical system that
Bohmian mechanics defines, much as the Maxwellian velocity
distribution emerges from an analysis of classical thermodynamic
equilibrium. (For more on the thermodynamic side of the analogy see
Goldstein 2001.) Thus with Bohmian mechanics the statistical
description in quantum theory indeed takes, as Einstein anticipated,
“an approximately analogous position to the statistical
mechanics within the framework of classical mechanics”.

10. Quantum Observables

Orthodox quantum theory supplies us with probabilities not merely for
positions but for a huge class of quantum observables. It might thus
appear that it is a much richer theory than Bohmian mechanics, which
seems exclusively concerned with positions. Appearances are, however,
misleading. In this regard, as with so much else in the foundations of
quantum mechanics, Bell made the crucial observation:

[I]n physics the only observations we must consider are position
observations, if only the positions of instrument pointers. It is a
great merit of the de Broglie-Bohm picture to force us to consider
this fact. If you make axioms, rather than definitions and theorems,
about the “measurement” of anything else, then you commit
redundancy and risk inconsistency. (Bell 1982, reprinted 1987c: 166)

Consider classical mechanics first. The observables are functions on
phase space, functions of the positions and momenta of the particles.
The axioms governing the behavior of the basic
observables—Newton’s equations for the positions or
Hamilton’s for positions and momenta—define the theory.
What would be the point of making additional axioms, for other
observables? After all, the behavior of the basic observables entirely
determines the behavior of any observable. For example, for classical
mechanics, the principle of the conservation of energy is a theorem,
not an axiom.

The situation might seem to differ in quantum mechanics, as usually
construed. Here there is no small set of basic observables having the
property that all other observables are functions of them. Moreover,
no observables at all are taken seriously as describing objective
properties, as actually having values whether or not they are or have
been measured. Rather, all talk of observables in quantum mechanics is
supposed to be understood as talk about the measurement of the
observables.

But if this is so, the situation with regard to other observables in
quantum mechanics is not really that different from that in classical
mechanics. Whatever quantum mechanics means by the measurement of (the
values of) observables—that, we are urged to believe,
don’t actually have values—must at least refer to some
experiment involving interaction between the “measured”
system and a “measuring” apparatus leading to a
recognizable result, as given potentially by, say, a pointer
orientation. But then if some axioms suffice for the behavior of
pointer orientations (at least when they are observed), rules about
the measurement of other observables must be theorems, following from
those axioms, not additional axioms.

It should be clear from the discussion
towards the end of Section 4
and at the
beginning of Section 9
that, assuming the quantum equilibrium hypothesis, any analysis of
the measurement of a quantum observable for orthodox quantum
theory—whatever it is taken to mean and however the
corresponding experiment is performed—provides ipso facto at
least as adequate an account for Bohmian mechanics. The only part of
orthodox quantum theory relevant to the analysis is the
Schrödinger evolution, and it shares this with Bohmian
mechanics. The main difference between them is that orthodox quantum
theory encounters
the measurement problem
before it reaches a satisfactory conclusion while Bohmian mechanics
does not. This difference stems of course from what Bohmian mechanics
adds to orthodox quantum theory: actual configurations.

The rest of this section will discuss the significance of quantum
observables for Bohmian mechanics. (It follows from what has been said
in the three preceding paragraphs that what we conclude here about
quantum observables for Bohmian mechanics holds for orthodox quantum
theory as well.)

Bohmian mechanics yields a natural association between experiments and
so-called generalized observables, given by
positive-operator-valued measures (Davies 1976), or POVMs, \(O(dz)\),
on the value spaces for the results of the experiments (Berndl,
Daumer, et al. 1995). This association is such that the probability
distribution of the result \(Z\) of an experiment, when performed upon
a system with wave function \(\psi\), is given by \(\langle \psi |
O(dz)\psi \rangle\) (where \(\langle | \rangle\) is the usual
inner product
between quantum state vectors).

Moreover, this conclusion is follows immediately from the very meaning
of an experiment from a Bohmian perspective: a coupling of system to
apparatus leading to a result \(Z\) that is a function of the final
configuration of the total system, e.g., the orientation of a pointer.
Analyzed in Bohmian mechanical terms, the experiment defines a map
from the initial wave function of the system to the distribution of
the result. It follows directly from the structure of Bohmian
mechanics, and from the fact that the quantum equilibrium distribution
is quadratic in the wave function, that this map is bilinear (or, more
precisely, sesquilinear, in that its dependence on one factor of the
wave function is antilinear, involving complex conjugation, rather
than linear). Such a map is equivalent to a POVM.

The simplest example of a POVM is a standard quantum observable,
corresponding to a self-adjoint operator \(A\) on the Hilbert space of
quantum states (i.e., wave functions). For Bohmian mechanics, more or
less every “measurement-like” experiment is associated
with this special kind of POVM. The familiar quantum measurement axiom
that the distribution of the result of the “measurement of the
observable \(A\)” is given by the spectral measure for
\(A\) relative to the wave function (in the very simplest cases just
the absolute squares of the so-called probability amplitudes)
is thus obtained.

For various reasons, after the discovery of quantum mechanics it
quickly became almost universal to speak of an experiment associated
with an operator \(A\) in the manner just sketched as a
measurement of the observable \(A\)—as if the
operator somehow corresponded to a property of the system that the
experiment in some sense measures. It has been argued that this
assumption, which has been called naive realism about
operators, has been a source of considerable confusion about the
meaning and implications of quantum theory (Daumer et al. 1997a).

11. Spin

The case of spin illustrates nicely both the way Bohmian mechanics
treats non-configurational quantum observables, and some of the
difficulties that the naive realism about operators mentioned above
causes.

Spin is the canonical quantum observable that has no classical
counterpart, reputedly impossible to grasp in a nonquantum way. The
difficulty is not quite that spin is quantized in the sense that its
allowable values form a discrete set (for a spin-1/2 particle, \(\pm
\hslash/2\)). Energy too may be quantized in this sense. Nor is it
precisely that the components of spin in the different directions fail
to commute—and so cannot be simultaneously discussed, measured,
imagined, or whatever it is that we are advised not to do with
noncommuting observables. Rather the problem is that there is no
ordinary (nonquantum) quantity which, like the spin observable, is a
3-vector and which also is such that its components in all possible
directions belong to the same discrete set. The problem, in other
words, is that the usual vector relationships among the various
components of the spin vector are incompatible with the quantization
conditions on the values of these components.

For a particle of spin-1 the problem is even more severe. The
components of spin in different directions aren’t simultaneously
measurable. Thus, the impossible vector relationships for the spin
components of a quantum particle are not observable. Bell (1966), and,
independently, Simon Kochen and Ernst Specker (1967) showed that for a
spin-1 particle the squares of the spin components in the various
directions satisfy, according to quantum theory, a collection of
relationships, each individually observable, that taken together are
impossible: the relationships are incompatible with the idea that
measurements of these observables merely reveal their preexisting
values rather than creating them, as quantum theory urges us to
believe. Many physicists and philosophers of physics continue to
regard the
Kochen-Specker Theorem
as precluding the possibility of hidden variables.

We thus might naturally wonder how Bohmian mechanics copes with spin.
But we have already answered this question. Bohmian mechanics makes
sense for particles with spin, i.e., for particles whose wave
functions are spinor-valued. When such particles are suitably directed
toward Stern-Gerlach magnets, they emerge moving in more or less a
discrete set of directions—2 possible directions for a spin-1/2
particle, having 2 spin components, 3 for spin-1 with 3 spin
components, and so on. This occurs because the Stern-Gerlach magnets
are so designed and oriented that a wave packet (a localized wave
function with reasonably well defined velocity) directed towards the
magnet will, by virtue of the Schrödinger evolution, separate
into distinct packets—corresponding to the spin components of
the wave function and moving in the discrete set of directions. The
particle itself, depending upon its initial position, ends up in one
of the packets moving in one of the directions.

The probability distribution for the result of such a
Stern-Gerlach experiment
can be conveniently expressed in terms of the quantum mechanical spin
operators—for a spin-1/2 particle given by certain 2 by 2
matrices called the Pauli spin matrices—in the manner alluded to
above.
From a Bohmian perspective there is no hint of paradox in any of
this—unless we assume that the spin operators correspond to
genuine properties of the particles.

For further discussion and more detailed examples of the Bohmian
perspective on spin see Norsen 2014.

12. Contextuality

The
Kochen-Specker Theorem,
the earlier theorem of Gleason (Gleason 1957 and Bell 1966), and
other no-hidden-variables results, including Bell’s inequality
(Bell 1964), show that any hidden-variables formulation of quantum
mechanics must be contextual. It must violate the
noncontextuality assumption “that measurement of an observable
must yield the same value independently of what other measurements may
be made simultaneously” (1964, reprinted 1987c: 9). To many
physicists and philosophers of science contextuality seems too great a
price to pay for the rather modest benefits—largely
psychological, so they would say—that hidden variables provide.

Even many Bohmians suggest that contextuality departs significantly
from classical principles. For example, Bohm and Hiley write that

The context dependence of results of measurements is a further
indication of how our interpretation does not imply a simple return to
the basic principles of classical physics. (1993: 100)

However, to understand contextuality in Bohmian mechanics almost
nothing needs to be explained. Consider an operator \(A\) that
commutes with operators \(B\) and \(C\) (which however don’t
commute with each other). What is often called the “result for
\(A\)” in an experiment for “measuring \(A\) together with
\(B\)” usually disagrees with the “result for \(A\)”
in an experiment for “measuring \(A\) together with
\(C\)”. This is because these experiments differ and different
experiments usually have different results. The misleading reference
to measurement, which suggests that a pre-existing value of \(A\) is
being revealed, makes contextuality seem more than it is.

Seen properly, contextuality amounts to little more than the rather
unremarkable observation that results of experiments should depend
upon how they are performed, even when the experiments are associated
with the same operator in the manner alluded to
above.
David Albert (1992: 153) has given a particularly simple and striking
example of this dependence for Stern-Gerlach experiments
“measuring” the \(z\)-component of spin. Reversing the
polarity in a magnet for “measuring” the \(z\)-component
of spin while keeping the same geometry yields another magnet for
“measuring” the \(z\)-component of spin. The use of one or
the other of these two magnets will often lead to opposite conclusions
about the “value of the \(z\)-component of spin” prior to
the “measurement” (for the same initial value of the
position of the particle).

As Bell insists:

A final moral concerns terminology. Why did such serious people take
so seriously axioms which now seem so arbitrary? I suspect that they
were misled by the pernicious misuse of the word
“measurement” in contemporary theory. This word very
strongly suggests the ascertaining of some preexisting property of
some thing, any instrument involved playing a purely passive role.
Quantum experiments are just not like that, as we learned especially
from Bohr. The results have to be regarded as the joint product of
“system” and “apparatus”, the complete
experimental set-up. But the misuse of the word
“measurement” makes it easy to forget this and then to
expect that the “results of measurements” should obey some
simple logic in which the apparatus is not mentioned. The resulting
difficulties soon show that any such logic is not ordinary logic. It
is my impression that the whole vast subject of “Quantum
Logic” has arisen in this way from the misuse of a word. I am
convinced that the word “measurement” has now been so
abused that the field would be significantly advanced by banning its
use altogether, in favour for example of the word
“experiment”. (Bell 1982, reprinted 1987c: 166)

13. Nonlocality

Bohmian mechanics is manifestly nonlocal. The velocity, as expressed
in
the guiding equation,
of any particle of a many-particle system will typically depend upon
the positions of the other, possibly distant, particles whenever the
wave function of the system is entangled, i.e., not a product of
single-particle wave functions. This is true, for example, for the
EPR-Bohm wave function, describing a pair of spin-1/2 particles in the
singlet state, that Bell and many others analyzed. Thus Bohmian
mechanics makes explicit the most dramatic feature of quantum theory:
quantum nonlocality, as discussed in Section 2.

It should be emphasized that the nonlocality of Bohmian mechanics
derives solely from the nonlocality, discussed in
Section 2,
built into the structure of standard quantum theory. This nonlocality
originates from a wave function on configuration space, an abstraction
which, roughly speaking, combines—or binds—distant
particles into a single irreducible reality. As Bell has stressed,

That the guiding wave, in the general case, propagates not in ordinary
three-space but in a multidimensional-configuration space is the
origin of the notorious “nonlocality” of quantum
mechanics. It is a merit of the de Broglie-Bohm version to bring this
out so explicitly that it cannot be ignored. (Bell 1980, reprinted
1987c: 115)

Thus the nonlocal velocity relation in the guiding equation is but one
aspect of the nonlocality of Bohmian mechanics. There is also the
nonlocality, or nonseparability, implicit in the wave function itself,
which is present even without the structure—actual
configurations—that Bohmian mechanics adds to orthodox quantum
theory. As Bell has shown, using the connection between the wave
function and the predictions of quantum theory about experimental
results, this nonlocality cannot easily be eliminated (see
Section 2).

The nonlocality of Bohmian mechanics can be appreciated perhaps most
efficiently, in all its aspects, by focusing on the
conditional wave function.
Suppose, for example, that in an EPR-Bohm experiment particle 1
passes through its Stern-Gerlach magnet before particle 2 arrives at
its magnet. Then the orientation of the Stern-Gerlach magnet for
particle 1 will significantly affect the conditional wave function of
particle 2: If the Stern-Gerlach magnet for particle 1 is oriented so
as to “measure the \(z\)-component of spin”, then after
particle 1 has passed through its magnet the conditional wave function
of particle 2 will be an
eigenvector
(or eigenstate) of the \(z\)-component of spin (in fact, belonging to
the eigenvalue that is the negative of the one “measured”
for particle 1), and the same thing is true for any other component of
spin. You can dictate the kind of spin eigenstate produced
for particle 2 by appropriately choosing the orientation of an
arbitrarily distant magnet. As to the future behavior of particle 2,
in particular how its magnet affects it, this of course depends very
much on the character of its conditional wave function and hence the
choice of orientation of the distant magnet strongly influences
it.

This nonlocal effect upon the conditional wave function of particle 2
follows from combining the standard analysis of the evolution of the
wave function in the EPR-Bohm experiment with the definition of the
conditional wave function. (For simplicity, we ignore permutation
symmetry.) Before reaching any magnets the EPR-Bohm wave function is a
sum of two terms, corresponding to nonvanishing values for two of the
four possible joint spin components for the two particles. Each term
is a product of an eigenstate for a component of spin in a given
direction for particle 1 with the opposite eigenstate (i.e., belonging
to the eigenvalue that is the negative of the eigenvalue for particle
1) for the component of spin in the same direction for particle 2.
Moreover, by virtue of its symmetry under rotations, the EPR-Bohm wave
function has the property that any component of spin, i.e., any
direction, can be used in this decomposition. (This property is very
interesting.)

Decomposing the EPR-Bohm wave function using the component of spin in
the direction associated with the magnet for particle 1, the evolution
of the wave function as particle 1 passes its magnet is easily
grasped: The evolution of the sum is determined (using the
linearity of Schrödinger’s equation)
by that of its individual terms, and the evolution of each term by
that of each of its factors. The evolution of the particle-1 factor
leads to a displacement along the magnetic axis in the direction
determined by the (sign of the) spin component (i.e., the eigenvalue),
as described in the fourth paragraph of
Section 11.
Once this displacement has occurred (and is large enough) the
conditional wave function for particle 2 will correspond to the term
in the sum selected by the actual position of particle 1. In
particular, it will be an eigenstate of the component of spin
“measured by” the magnet for particle 1. (For a more
explicit and detailed discussion see Norsen 2014.)

The nonlocality of Bohmian mechanics has a remarkable feature: it is
screened by quantum equilibrium. It is a consequence of the
quantum equilibrium hypothesis
that the nonlocal effects in Bohmian mechanics don’t yield
observable consequences that can be controlled—we can’t
use them to send instantaneous messages. This follows from the fact
that, given the quantum equilibrium hypothesis, the observable
consequences of Bohmian mechanics are the same as those of orthodox
quantum theory, for which instantaneous communication based on quantum
nonlocality is impossible (see Eberhard 1978). Valentini (1991)
emphasizes the importance of quantum equilibrium for obscuring the
nonlocality of Bohmian mechanics. (Valentini [2010a] has also
suggested the possibility of searching for and exploiting quantum
non-equilibrium. However, in contrast with thermodynamic
non-equilibrium, we have at present no idea what quantum
non-equilibrium, should it exist, would look like, despite claims and
arguments to the contrary.)

14. Lorentz Invariance

Like nonrelativistic quantum theory, of which it is a version, Bohmian
mechanics and special relativity, a central principle of physics, are
not compatible: Bohmian mechanics is not Lorentz invariant. Nor can it
easily be modified to accommodate Lorentz invariance. Configurations,
defined by the simultaneous positions of all particles, play
too crucial a role in its formulation, with
the guiding equation
defining an evolution on configuration space. (Lorentz
invariant extensions of Bohmian mechanics for a single particle,
described by the Dirac equation (Bohm & Hiley 1993; Dürr et
al. 1999) or the Klein-Gordon equation (Berndl et al. 1996; Nikolic
2005), can easily be achieved, though for a Klein-Gordon particle
there are some interesting subtleties, corresponding to what might
seem to be a particle traveling backwards in time.)

This difficulty with Lorentz invariance and the nonlocality in Bohmian
mechanics are closely related. Since quantum theory itself, by virtue
merely of the character of its predictions concerning EPR-Bohm
correlations, is irreducibly nonlocal (see
Section 2),
one might expect considerable difficulty with the Lorentz invariance
of orthodox quantum theory as well with Bohmian mechanics. For
example, the collapse rule of textbook quantum theory blatantly
violates Lorentz invariance. As a matter of fact, the intrinsic
nonlocality of quantum theory presents formidable difficulties for the
development of any (many-particle) Lorentz invariant formulation that
avoids the vagueness of orthodox quantum theory (see Maudlin
1994).

Bell made a somewhat surprising evaluation of the importance of the
problem of Lorentz invariance. In an interview with the philosopher
Renée Weber, not long before he died, he referred to the
paradoxes of quantum mechanics and observed that

Those paradoxes are simply disposed of by the 1952 theory of Bohm,
leaving as the question, the question of Lorentz invariance.
So one of my missions in life is to get people to see that if they
want to talk about the problems of quantum mechanics—the real
problems of quantum mechanics—they must be talking about Lorentz
invariance. (Interview with John Bell, in Weber 1989, Other Internet
Resources)

The most common view on this issue is that a detailed description of
microscopic quantum processes, such as would be provided by a putative
extension of Bohmian mechanics to the relativistic domain, must
violate Lorentz invariance. In this view Lorentz invariance in such a
theory would be an emergent symmetry obeyed by our
observations—for Bohmian mechanics a statistical consequence of
quantum equilibrium that governs the results of quantum experiments.
This is the opinion of Bohm and Hiley (1993), of Holland (1993), and
of Valentini (1997).

However—unlike nonlocality—violating Lorentz invariance is
not inevitable. It should be possible, it seems, to construct a fully
Lorentz invariant theory that provides a detailed description of
microscopic quantum processes. One way to do this is by using an
additional Lorentz invariant dynamical structure, for example a
suitable time-like 4-vector field, that permits the definition of a
foliation of space-time into space-like hypersurfaces providing a
Lorentz invariant notion of “evolving configuration” and
along which nonlocal effects are transmitted. See Dürr et al.
1999 for a toy model. Another possibility is that a fully Lorentz
invariant account of quantum nonlocality can be achieved without the
invocation of additional structure, exploiting only what is already at
hand, for example, the wave function of the universe or light-cone
structure. For more on the latter possibility, see Goldstein and
Tumulka’s model (2003), in which they reconcile relativity and
nonlocality through the interplay of opposite arrows of time. For a
discussion of the former possibility, see Dürr et al. 2014. In
the sort of theory discussed there, the wave function of the universe
provides a covariant prescription for the desired foliation. Such a
theory would be clearly Lorentz invariant. But it is not so clear that
it should be regarded as relativistic.

Be that as it may, Lorentz invariant nonlocality remains somewhat
enigmatic. The issues are extremely subtle. For example, Bell rightly
would find

disturbing … the impossibility of “messages” faster
than light, which follows from ordinary relativistic quantum mechanics
in so far as it is unambiguous and adequate for procedures \(we\)
[emphasis added] can actually perform. The exact elucidation of
concepts like “message” and “we”, would be a
formidable challenge. (1981a, reprinted 1987c: 155)

While quantum equilibrium and the absolute uncertainty that it entails
(Dürr, Goldstein, & Zanghì 1992a) may be of some help
here, the situation remains puzzling.

15. Objections and Responses

Bohmian mechanics has never been widely accepted in the mainstream of
the physics community. Since it is not part of the standard physics
curriculum, many physicists—probably the majority—are
simply unfamiliar with the theory and how it works. Sometimes the
theory is rejected without explicit discussion of reasons for
rejection. One also finds objections that are based on simple
misunderstandings; among these are claims that some no-go theorem,
such as von Neumann’s theorem, the Kochen-Specker theorem, or
Bell’s theorem, shows that the theory cannot work. Such
objections will not be dealt with here, as the reply to them will be
obvious to those who understand the theory. In what follows only
objections that are not based on elementary misunderstandings will be
discussed.

A common objection is that Bohmian mechanics is too complicated or
inelegant. To evaluate this objection one must compare the axioms of
Bohmian mechanics with those of standard quantum mechanics. To
Schrödinger’s equation, Bohmian mechanics adds the
guiding equation;
standard quantum mechanics instead requires postulates about
experimental outcomes that can only be formulated in terms of a
distinction between a quantum system and the experimental apparatus.
And, as noted by Hilary Putnam,

In Putnam ([1965]), I rejected Bohm’s interpretation for several
reasons which no longer seem good to me. Even today, if you look at
the Wikipedia encyclopaedia on the Web, you will find it said that
Bohm’s theory is mathematically inelegant. Happily, I did not
give that reason in Putnam ([1965]), but in any case it is
not true. The formula for the velocity field is extremely simple: you
have the probability current in the theory anyway, and you take the
velocity vector to be proportional to the current. There is nothing
particularly inelegant about that; if anything, it is remarkably
elegant! (2005: 262)

One frequent objection is that Bohmian mechanics, since it makes
precisely the same predictions as standard quantum mechanics (insofar
as the predictions of standard quantum mechanics are unambiguous), is
not a distinct theory but merely a reformulation of standard quantum
theory. In this vein, Heisenberg wrote,

Bohm’s interpretation cannot be refuted by experiment, and this
is true of all the counter-proposals in the first group. From the
fundamentally “positivistic” (it would perhaps be better
to say “purely physical”) standpoint, we are thus
concerned not with counter-proposals to the Copenhagen interpretation,
but with its exact repetition in a different language. (Heisenberg
1955: 18)

More recently, Sir Anthony Leggett has echoed this charge. Referring
to the measurement problem, he says that Bohmian mechanics provides
“little more than verbal window dressing of the basic
paradox” (Leggett 2005: 871). And in connection with the
double-slit experiment, he writes,

No experimental consequences are drawn from [the assumption of
definite particle trajectories] other than the standard predictions of
the QM formalism, so whether one regards it as a substantive
resolution of the apparent paradox or as little more than a
reformulation of it is no doubt a matter of personal taste (the
present author inclines towards the latter point of view). (Leggett
2002: R419)

Now Bohmian mechanics and standard quantum mechanics provide clearly
different descriptions of what is happening on the microscopic quantum
level. So it is only with a purely instrumental attitude towards
scientific theories that Bohmian mechanics and standard quantum
mechanics can possibly be regarded as different formulations of
exactly the same theory. But even if they were, why would this be an
objection to Bohmian mechanics? Even if they were, we should still ask
which of the two formulations is superior. Those impressed by the
“not-a-distinct-theory” objection presumably give
considerable weight to the fact that standard quantum mechanics came
first. Supporters of Bohmian mechanics give more weight to its greater
simplicity and clarity.

The position of Leggett, however, is very difficult to understand.
There should be no measurement problem for a physicist with a purely
instrumentalist understanding of quantum mechanics. But for more than
thirty years Leggett has forcefully argued that quantum mechanics
indeed suffers from the measurement problem. For Leggett the problem
is so serious that it has led him to suggest that quantum mechanics
might fail on the macroscopic level. Thus Leggett is no
instrumentalist, and it is hard to understand why he so cavalierly
dismisses a theory like Bohmian mechanics that obviously doesn’t
suffer from the measurement problem, with which he has been so long
concerned.

Sir Roger Penrose also seems to have doubts as to whether Bohmian
mechanics indeed resolves the measurement problem. He writes that

it seems to me that some measure of scale is indeed needed, for
defining when classical-like behaviour begins to take over from
small-scale quantum activity. In common with the other quantum
ontologies in which no measurable deviations from standard quantum
mechanics is expected, the point of view (e) [Bohmian mechanics] does
not possess such a scale measure, so I do not see that it can
adequately address the paradox of Schrödinger’s cat. (2005:
811)

But contrary to what he writes, his real concern seems to be with the
emergence of classical behavior, and not with the measurement problem
per se. With regard to this, we note that the Bohmian
evolution of particles, which is always governed by the wave function
and is always fundamentally quantum, turns out to be approximately
classical when the relevant de Broglie wave length, determined in part
by the wave function, is much smaller than the scale on which the
potential energy term in Schrödinger’s equation varies (see
Allori et al., 2002). Under normal circumstances this condition will
be satisfied for the center of mass motion of a macroscopic
object.

It is perhaps worth mentioning that despite the empirical equivalence
between Bohmian mechanics and orthodox quantum theory, there are a
variety of experiments and experimental issues that don’t fit
comfortably within the standard quantum formalism but are easily
handled by Bohmian mechanics. Among these are dwell and tunneling
times (Leavens 1996), escape times and escape positions (Daumer et al.
1997b), scattering theory (Dürr et al., 2000), and quantum chaos
(Cushing 1994; Dürr, Goldstein, & Zanghì 1992b).
Especially problematical from an orthodox perspective is quantum
cosmology, for which the relevant quantum system is the entire
universe, and hence there is no observer outside the system to cause
collapse of the wave function upon measurement. In this setting
Bohmian models have clarified the issue of the inevitability of the
presence of singularities in theories of quantum gravity (Falciano,
Pinto-Neto, & Struyve 2015). Moreover, the additional resources
present in Bohmian mechanics, in particular the notion of the
conditional wave function defined in Section 8, have been useful for
the development of approximation schemes for dealing with practical
quantum applications (Oriols & Mompart 2012).

Another claim that has become popular in recent years is that Bohmian
mechanics is an Everettian, or “many worlds”,
interpretation in disguise (see entry on
the many worlds interpretation of quantum mechanics
for an overview of such interpretations). The idea is that Bohmians,
like Everettians, must take the wave-function as physically real.
Moreover, since Bohmian mechanics involves no wave-function collapse
(for the wave function of the universe), all of the branches of the
wave function, and not just the one that happens to be occupied by the
actual particle configuration, persist. These branches are those that
Everettians regard as representing parallel worlds. As David Deutsch
expresses the charge,

the “unoccupied grooves” must be physically real. Moreover
they obey the same laws of physics as the “occupied
groove” that is supposed to be “the” universe. But
that is just another way of saying that they are universes too.
… In short, pilot-wave theories are parallel-universes theories
in a state of chronic denial. (Deutsch 1996: 225)

See Brown and Wallace (2005) for an extended version of this argument.
Not surprisingly, Bohmians do not agree that the branches of the wave
function should be construed as representing worlds. For one Bohmian
response, see Maudlin (2010). Other Bohmian responses have been given
by Lewis (2007) and Valentini (2010b).

The claim of Deutsch, Brown, and Wallace is of a novel character that
we should perhaps pause to examine. On the one hand, for anyone who,
like Wallace, accepts the viability of a functionalist many-worlds
understanding of quantum mechanics—and in particular accepts
that it follows as a matter of functional and structural analysis that
when the wave function develops suitable complex patterns these ipso
facto describe what we should regard as worlds—the claim should
be compelling. On the other hand, for those who reject the functional
analysis and regard many worlds as ontologically inadequate (see
Maudlin 2010), or who, like Vaidman (see the SEP entry on the
many-worlds interpretation of quantum mechanics),
accepts many worlds on non-functionalist grounds, the claim should
seem empty. In other words, one has basically to have already accepted
a strong version of many worlds and already rejected Bohm in order to
feel the force of the claim.

Another interesting aspect of the claim is this: It seems that one
could consider, at least as a logical possibility, a world consisting
of particles moving according to some well-defined equations of
motion, and in particular according to the equations of Bohmian
mechanics. It seems entirely implausible that there should be a
logical problem with doing so. We should be extremely sceptical of any
argument, like the claim of Deutsch, Brown, and Wallace, that suggests
that there is. Thus what, in defense of many worlds, Deutsch, Brown,
and Wallace present as an objection to Bohmian mechanics should
perhaps be regarded instead as an objection to many worlds itself.

There is one striking feature of Bohmian mechanics that is often
presented as an objection: in Bohmian mechanics the wave function acts
upon the positions of the particles but, evolving as it does
autonomously via Schrödinger’s equation, it is not acted
upon by the particles. This is regarded by some Bohmians, not as an
objectionable feature of the theory, but as an important clue about
the meaning of the quantum-mechanical wave function. Dürr,
Goldstein, & Zanghì (1997) and Goldstein & Teufel
(2001) discuss this point and suggest that from a deeper perspective
than afforded by standard Bohmian mechanics or quantum theory, the
wave function should be regarded as nomological, as an object for
conveniently expressing the law of motion somewhat analogous to the
Hamiltonian in classical mechanics, and that a time-dependent
Schrödinger-type equation, from this deeper (cosmological)
perspective, is merely phenomenological.

Bohmian mechanics does not account for phenomena such as particle
creation and annihilation characteristic of quantum field theory. This
is not an objection to Bohmian mechanics but merely a recognition that
quantum field theory explains a great deal more than does
nonrelativistic quantum mechanics, whether in orthodox or Bohmian
form. It does, however, underline the need to find an adequate, if not
compelling, Bohmian version of quantum field theory, and of gauge
theories in particular. Some rather tentative steps in this direction
can be found in Bohm & Hiley 1993, Holland 1993, Bell 1987b), and
in some of the articles in Cushing, Fine, & Goldstein 1996. A
crucial issue is whether a quantum field theory is fundamentally about
fields or particles—or something else entirely. While the most
common choice is fields (see Struyve 2010 for an assessment of a
variety of possibilities), Bell’s is particles. His proposal is
in fact the basis of a canonical extension of Bohmian mechanics to
general quantum field theories, and these “Bell-type quantum
field theories” (Dürr et al. 2004 and 2005) describe a
stochastic evolution of particles that involves particle creation and
annihilation. (For a general discussion of this issue, and of the
point and value of Bohmian mechanics, see the exchange of letters
between Goldstein and Weinberg by following the link provided in the
Other Internet Resources
section below.)

For an accessible introduction to Bohmian mechanics see Bricmont
2016.

Heisenberg, Werner, 1955, “The Development of the
Interpretation of the Quantum Theory”, in W. Pauli (ed.),
Niels Bohr and the Development of Physics: Essays Dedicated to
Niels Bohr on the Occasion of his Seventieth Birthday, New York:
McGraw-Hill, pp. 12–29.